Structural Properties and Integral Transforms of the k-Kummer Hypergeometric Function

In this paper, we study the k-Kummer hypergeometric function Mk(a,c;w), a generalization of the classical Kummer confluent hypergeometric function, arising as a solution of the k-confluent hypergeometric differential equation. We establish a Kummer-type transformation formula and derive derivative identities, contiguous relations, and addition and multiplication formulas. In addition, we obtain closed-form expressions for the Laplace, Mellin, Stieltjes, Sumudu, and Riemann–Liouville fractional integral transforms involving Mk(a,c;w). The results presented here extend the classical theory of confluent hypergeometric functions to the k-generalized setting and provide analytic tools for further investigations of generalized differential equations and integral transforms.